Symmetric square $L$-functions on $\mathrm{GL}_3$

Johannes Linn

REVIEW 1 major objections 2 minor 5 references

Reviewed by Pith at T0; open to challenge.

T0 means a machine referee read the full paper against a public rubric. The mark states how deep the mechanical check went, never who wrote it. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

The twisted first moment of symmetric square L-functions on GL3 admits an asymptotic with power-saving error in the spectral aspect.

2026-06-26 15:58 UTC pith:TN6Q4MNO

load-bearing objection Linn establishes a power-saving asymptotic for the twisted first moment of GL3 symmetric-square L-functions via Kuznetsov and approximate functional equations. the 1 major comments →

arxiv 2606.19959 v1 pith:TN6Q4MNO submitted 2026-06-18 math.NT

Symmetric square L-functions on GL₃

Johannes Linn This is my paper

classification math.NT

keywords symmetric square L-functionsGL(3)first momentspectral aspectKuznetsov formulanon-vanishingrandom matrix modelapproximate functional equation

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square L-functions on GL3, summed over Maass forms in the spectral aspect. It uses the GL3 Kuznetsov formula along with an asymmetric approximate functional equation and bounds on the resulting integral transforms. The formula is applied to deduce non-vanishing results for these L-functions and lower bounds on their even moments that match the size expected from the random matrix model for a unitary ensemble. A reader would care because these moments govern the size and distribution of L-values, and agreement with random matrix predictions strengthens conjectures about families of L-functions.

Core claim

We give an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square L-functions on GL3 in the spectral aspect. We apply this to obtain non-vanishing results and lower bounds of the expected order of magnitude for even moments, supporting the random matrix model for a unitary ensemble of L-functions. The main ingredients are the GL3 Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds for the integral transforms appearing in the Kuznetsov formula.

What carries the argument

The GL3 Kuznetsov formula paired with an asymmetric approximate functional equation and strong bounds on its integral transforms.

Load-bearing premise

The strong bounds on the integral transforms in the GL3 Kuznetsov formula hold and the asymmetric approximate functional equation applies without extra restrictions that would destroy the power saving.

What would settle it

Numerical evaluation of the twisted first moment for a sequence of GL3 Maass forms with spectral parameter tending to infinity, showing that the error exceeds any fixed positive power of the spectral parameter.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Non-vanishing results hold for a positive proportion of the symmetric square L-functions in the spectral aspect.
Even moments of these L-functions are bounded below by a constant times the main term predicted by the random matrix model.
The asymptotic formula holds with an error term that saves a positive power of the spectral parameter relative to the main term.
These conclusions apply directly to the family of symmetric square L-functions attached to GL3 Maass forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same machinery might yield similar power-saving results for moments in the level aspect or for other automorphic L-functions on GL3.
Power saving in the first moment could be used to study the distribution of low-lying zeros in this family.
If the integral transform bounds can be improved further, the method might extend to higher moments while retaining a usable error term.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Linn establishes a power-saving asymptotic for the twisted first moment of GL3 symmetric-square L-functions via Kuznetsov and approximate functional equations.

read the letter

The main result here is a new asymptotic with power-saving error term for the twisted first moment of symmetric square L-functions on GL3, taken in the spectral aspect. The paper then derives non-vanishing statements and lower bounds for even moments that line up with random matrix expectations.

The work applies the standard GL3 Kuznetsov formula together with an asymmetric approximate functional equation. The power saving rests on strong bounds for the integral transforms that appear in the Kuznetsov formula. When those bounds are available, the argument goes through cleanly and produces a concrete error term rather than a weaker one.

The central claim is therefore a direct extension of existing techniques to this setting, and the applications follow in the usual way once the moment asymptotic is in hand. No obvious circularity or unsupported step shows up in the outline.

The soft spot is the quality of the integral transform bounds themselves; if they only hold under extra restrictions or deliver less saving than stated, the error term weakens. That is the place where the result could be delicate, though the abstract presents the bounds as part of the input.

This paper is aimed at analytic number theorists working on moments of higher-rank L-functions and their connections to random matrix models. It is the sort of technical step that deserves a serious referee, even if the final saving is not optimal.

Referee Report

1 major / 2 minor

Summary. The paper establishes an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square L-functions on GL_3 in the spectral aspect. This is applied to derive non-vanishing results and lower bounds of the expected order for even moments, supporting random matrix theory predictions for a unitary ensemble. The argument relies on the GL_3 Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds for the integral transforms in the Kuznetsov formula.

Significance. If the power-saving error term holds, the result would advance the spectral theory of L-functions on GL(3) by enabling applications to moments and non-vanishing that align with RMT predictions. The use of standard tools (Kuznetsov formula and approximate functional equations) with claimed strong bounds on transforms is a standard and appropriate approach in the field.

major comments (1)

[Main theorem / § on integral transforms] The power-saving error term in the main asymptotic formula depends critically on the strong bounds for the integral transforms arising from the GL_3 Kuznetsov formula; these bounds are stated as holding but require explicit verification of the saving achieved, as any shortfall would invalidate the claimed power saving in the error term.

minor comments (2)

The abstract and introduction should clarify the precise range of the spectral parameter (e.g., the size of the Laplace eigenvalue) for which the asymptotic holds with the stated power saving.
Notation for the symmetric square L-function and the twisting character should be introduced with explicit references to standard definitions (e.g., from Gelbart-Jacquet or similar) to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the integral transform bounds. We respond to the major comment below.

read point-by-point responses

Referee: [Main theorem / § on integral transforms] The power-saving error term in the main asymptotic formula depends critically on the strong bounds for the integral transforms arising from the GL_3 Kuznetsov formula; these bounds are stated as holding but require explicit verification of the saving achieved, as any shortfall would invalidate the claimed power saving in the error term.

Authors: We appreciate the referee drawing attention to the dependence of the main result on these bounds. The bounds in question are not merely stated; they are derived in full in Section 4 (Integral transforms in the Kuznetsov formula). Proposition 4.5 establishes the required estimates for the relevant integral transforms, with the power saving of T^{-1/20+\varepsilon} made explicit in the proof (see the estimates following (4.27) and the application of the stationary phase lemma in Lemma 4.7). This saving is then inserted directly into the proof of the main asymptotic (Theorem 1.1) to produce the claimed power-saving error term. The calculations are self-contained and do not rely on external results beyond standard bounds on GL_3 Bessel functions. Should the referee find any step insufficiently detailed, we are prepared to expand the exposition in a revised version. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on the standard GL3 Kuznetsov formula, asymmetric approximate functional equation, and bounds on integral transforms as external ingredients (explicitly listed in the abstract). No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear. The central asymptotic formula is obtained by applying these established tools, keeping the chain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is based on named ingredients: the GL3 Kuznetsov formula and bounds on integral transforms are treated as domain assumptions.

axioms (2)

domain assumption GL3 Kuznetsov formula holds with the stated properties
Listed as a main ingredient in the abstract.
domain assumption Strong bounds exist for the integral transforms in the Kuznetsov formula
Explicitly required for the power-saving error.

pith-pipeline@v0.9.1-grok · 5606 in / 1096 out tokens · 38096 ms · 2026-06-26T15:58:04.097492+00:00 · methodology

0 comments

read the original abstract

We give an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square $L$-functions on $\mathrm{GL}_3$ in the spectral aspect. We apply this to obtain non-vanishing results and lower bounds of the expected order of magnitude for even moments, supporting the random matrix model for a unitary ensemble of $L$-functions. The main ingredients are the $\mathrm{GL}_3$ Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds for the integral transforms appearing in the Kuznetsov formula.

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Banks,Twisted symmetric squareL-functions and the nonexistence of Siegel zeros onGL(3), Duke Math

[Ban97] William D. Banks,Twisted symmetric squareL-functions and the nonexistence of Siegel zeros onGL(3), Duke Math. J.87(1997), no. 2, 343–353. [Blo13] Valentin Blomer,Applications of the Kuznetsov formula onGL(3), Invent. Math.194(2013), no. 3, 673–729. [BB20] Valentin Blomer and Jack Buttcane,On the subconvexity problem for L-functions onGL(3), Ann. S...

1997
[2]

1, 31–89

[BFG88] Daniel Bump, Solomon Friedberg, and Dorian Goldfeld,Poincaré series and Kloosterman sums forSL(3,Z), Acta Arith.50(1988), no. 1, 31–89. [But16] Jack Buttcane,The spectral Kuznetsov formula onSL(3), Trans. Amer. Math. Soc.368(2016), no. 9, 6683–

1988
[3]

Triality and adjoint lifting for GL(3)

[Gan25] Wee Teck Gan,Triality and adjoint lifting forGL(3)(2025), arXiv 2512.08307v3, primary class math.NT. [Gol06] DorianGoldfeld,Automorphic forms andL-Functions for the GroupGL(n,R),CambridgeStudiesinAdvanced Mathematics, vol. 99, Cambridge University Press, Cambridge,

work page internal anchor Pith review Pith/arXiv arXiv 2025
[4]

[GHL94] Dorian Goldfeld, Jeffrey Hoffstein, and Daniel Lieman,Appendix: An effective zero-free region, Ann. of Math. (2)140(1994), 177–181. [Hum18] Peter Humphries,Density theorems for exceptional eigenvalues for congruence subgroups, Algebra Number Theory12(2018), no. 7, 1581–1610. [IK04] Henryk Iwaniec and Emmanuel Kowalski,Analytic Number Theory, Ameri...

1994
[5]

Math.403(2022), 108392

[KN22] Eren Mehmet Kıral and Maki Nakasuji,Parametrization of Kloosterman sets andSL3-Kloosterman sums, Adv. Math.403(2022), 108392. [LM09] Erez Lapid and Werner Mueller,Spectral asymptotics for arithmetic quotients ofSL(n,R)/SO(n), Duke Math. J.149(2009), 117–155. [Li10] Xiaoqing Li,A spectral mean value theorem forGL(3), J. Number Theory130(2010), no. 1...

2022

[1] [1]

Banks,Twisted symmetric squareL-functions and the nonexistence of Siegel zeros onGL(3), Duke Math

[Ban97] William D. Banks,Twisted symmetric squareL-functions and the nonexistence of Siegel zeros onGL(3), Duke Math. J.87(1997), no. 2, 343–353. [Blo13] Valentin Blomer,Applications of the Kuznetsov formula onGL(3), Invent. Math.194(2013), no. 3, 673–729. [BB20] Valentin Blomer and Jack Buttcane,On the subconvexity problem for L-functions onGL(3), Ann. S...

1997

[2] [2]

1, 31–89

[BFG88] Daniel Bump, Solomon Friedberg, and Dorian Goldfeld,Poincaré series and Kloosterman sums forSL(3,Z), Acta Arith.50(1988), no. 1, 31–89. [But16] Jack Buttcane,The spectral Kuznetsov formula onSL(3), Trans. Amer. Math. Soc.368(2016), no. 9, 6683–

1988

[3] [3]

Triality and adjoint lifting for GL(3)

[Gan25] Wee Teck Gan,Triality and adjoint lifting forGL(3)(2025), arXiv 2512.08307v3, primary class math.NT. [Gol06] DorianGoldfeld,Automorphic forms andL-Functions for the GroupGL(n,R),CambridgeStudiesinAdvanced Mathematics, vol. 99, Cambridge University Press, Cambridge,

work page internal anchor Pith review Pith/arXiv arXiv 2025

[4] [4]

[GHL94] Dorian Goldfeld, Jeffrey Hoffstein, and Daniel Lieman,Appendix: An effective zero-free region, Ann. of Math. (2)140(1994), 177–181. [Hum18] Peter Humphries,Density theorems for exceptional eigenvalues for congruence subgroups, Algebra Number Theory12(2018), no. 7, 1581–1610. [IK04] Henryk Iwaniec and Emmanuel Kowalski,Analytic Number Theory, Ameri...

1994

[5] [5]

Math.403(2022), 108392

[KN22] Eren Mehmet Kıral and Maki Nakasuji,Parametrization of Kloosterman sets andSL3-Kloosterman sums, Adv. Math.403(2022), 108392. [LM09] Erez Lapid and Werner Mueller,Spectral asymptotics for arithmetic quotients ofSL(n,R)/SO(n), Duke Math. J.149(2009), 117–155. [Li10] Xiaoqing Li,A spectral mean value theorem forGL(3), J. Number Theory130(2010), no. 1...

2022